# Topology Qualifying Exam Workshop

May 2020

Worksheet 2A

Theme: Homeomorphism invariants: Separation properties $( T _ { 0 } , T _ { 1 }$ , Hausdorff, regular, normal), countability properties; connectedness, path connectedness, components; compactness, metrizability.
Applications.

Part 1 - 5-10 minutes

Warm-up:

1. What are the properties $T _ { 0 } - T _ { 4 } ?$ What other names are there for $T _ { 2 } - T _ { 4 } ?$

2. Pick an $n \in \{ 0 , 1 , 2 , 3 \}$ , and give an example of a space that is $T _ { n }$ , but not $T _ { n + 1 }$

3. What property on a space guarantees that compact sets are closed?
   What property on a space guarantees that closed sets are compact?

4. Show that if there is a path between two points, they lie in the same connected component.
   Why does this show that a path connected space is connected?

5. Optional: Explain why the “topologist’s sine wave” $X = \{ ( 0 , 0 ) \} \cup \{ ( x , \sin ( 1 / x ) ) : x > 0 \}$ is an example of a space which is connected but not path connected.

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (May 2015) If $X = \mathbb { R } ^ { n }$ with the usual Euclidean topology and $\mathcal { U } \subseteq X$ is an open, connected subspace of X, show that U is also path connected.

2. (January 2017) A space X is completely regular if X is $T _ { 1 }$ and whenever C is closed in X and $p \in X \backslash C .$ then there is a continuous function $f : X \to [ 0 , 1 ]$ such that $f ( p ) = 0$ and $f ( C ) = \{ 1 \}$

(a) Let X be a $T _ { 1 }$ space with topology T generated by a subbasis S. Show that X is completely regular if and only if for every $p \in X$ and subbasis element $U \in S$ containing $p ,$ there is a continuous function $f : X \to [ 0 , 1 ]$ such that $f ( p ) = 0$ and $f ( X \setminus U ) = \{ 1 \}$

(b) Show that if X and Y are completely regular spaces, then so is the product space $X \times Y$ . (Hint: use the result in (a).)

3. (May 2018) Show that a closed subset $A \subseteq X$ of a normal (i.e. T1 and $T _ { 4 } )$ space $( X , \tau )$ , with the subspace topology, is normal.

4. (May 2019) Show that a topological space (X, T ) is Hausdorff if and only if the diagonal $\Delta = \{ ( x , x )$ $x \in X \}$ is a closed subset of $X \times X$ , where $X \times X$ is equipped with the product topology.

5. (May 2019) Recall that a topological space X is second countable if there exists a countable basis for the topology on X, and X is separable if it has a countable dense subset.

(a) Show that if $( X , \tau )$ has a metric topology, from the metric d : $X \times X \to \mathbb { R }$ , and X is separable, then X is second countable.

(b) Show that R, with the lower limit topology $\mathcal { T } _ { \ell } ^ { 1 }$ is not metrizable, i.e., T\` is not the topology from any metric on R.

Part 3 - 5-10 minutes Wrap-up discussion with everyone:

1. Which of these problems did you find easy?
   Hard?

2. Which topics in the theme do you need to review more?